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DISCRETE

MATHEMATICS

ELSEVIER Discrete Mathematics 156 (1996) 243-246

Communication

On a Vizing-like conjecture for direct product graphs 1

Sandi Klav~ar*, Bla~ Zmazek

Department of Mathematics, PEF, University of Maribor, Koro~ka cesta 160, 2000 Maribor, Slovenia

Received 5 February 1996

Communicated by C. Benzaken

Abstract

Let 7(G) be the domination number of a graph G, and let G ×H be the direct product of graphs

G and H. It is shown that for any k t> 0 there exists a graph G such that 7(G × G) ~< 7(G) 2 -k.

This in particular disproves a conjecture from [5].

1. Introduction

A set D of vertices of a simple graph G is called dominating if every vertex

w E V(G)-D is adjacent to some vertex rED. The domination number of a graph

G, 7(G), is the order of a smallest dominating set of G. A dominating set D with

IDI -- v(G) is called a minimum dominating set.

The direct product G x H of graphs G and H is a graph with V(G × H) =

V(G) × V(H) and E(G x H)= {{(a,x),(b,y)}l{a,b} E E(G) and {x,y} E E(H)}.

This product is also known as Kronecker product, tensor product, categorical prod-

uct and graph conjunction. The Cartesian product G½H of graphs G and H is the

graph with vertex set V(G) × V(H) and (a,x)(b,y) E E(G[ZH) whenever x = y and

{a,b} E E(G), or a = b and {x,y} E E(H).

Most of the interest for domination in graph products is due to Vizing's conjec-

ture [11] from 1963. Vizing conjectured that

7(GVqH) t> 7(G)7(H)

hold for any graphs G and H. Despite considerable efforts (cf. [1-4,6-9]) it seems

that presently there is no 'winning way' to the conjecture.

* Corresponding author. E-mail: sandi.klavzar@uni-lj.si; blaz.zmazek@fmf.uni-lj.si.

1 This work was supported in part by the Ministry of Science and Technology of Slovenia under the grant

J1-7036.

0012-365X/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved

Pll S0012-365X(96)00032-5

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244 S. Klav~ar, B. Zmazek /Discrete Mathematics 156 (1996) 243-246

Another graph product which offers interesting and non-trivial problems on domi-

nation is the direct product. Gravier and Khelladi [5] posed the following Vizing-like

conjecture for the direct product:

?(G x H)/> ?(G)?(H).

Here we show that for any k/> 0 there exists a graph G such that 7(G× G) ~< 7(G)2-k.

This result in particular disproves the above-mentioned conjecture. Moreover, it also

supports the following statement: although the direct product of graphs is the most

natural graph product, it is also the most difficult and unpredictable among standard

graph products.

In fact, as far as we know Nowakowski and Rail were the first who observed that

the above-mentioned conjecture does not hold. In their manuscript [10] they report a

graph with 7(G) = 2 yet 7(G x G) -- 3. We wish to add that the paper of Nowakowski

and Rail is a nice and relevant paper which considers several graph parameters (related

to independence, domination and irredundance) of all main associative graph products.

2. The construction

Let G1 be the graph depicted in Fig. 1 and let H be the graph Gl\{U,W} (see Fig. 1

again). Then we have:

Lemma 2.1. (i) T(G1) -- y(H) = 3.

(ii) y(G1 x G1 ) ~< 7.

Proof. (i) Partition V(G1) into the sets V1 = {x,y,z,.f,.~,~.}, I1"2 = {u,v,w} and

V3 = {if, ,7, ~} and note that the domination number of the subgraph of G1 induced by

the set 1:1 is equal 2.

/-¢

Fig. 1. Graphs GI and H.

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S. Klav~ar, B. ZmazeklDiscrete Mathematics 156 (1996) 243-246

£ a?

...... ~(n)

Fig. 2. Graph Gn.

245

Suppose that 7(G1 ) = 2 and let D be a minimum dominating set. Since any pair of

vertices from the set {£, y,Z} have no common neighbour in V2, at least one vertex

of Vt must belong to D. Therefore, the other vertex of D must lie in V2. But this

means that at least one vertex of V3 is not dominated by D, a contradiction. Clearly,

y(G1) ~< 3.

Analogous argument (with I:2 = {v}) also gives 7(H) = 3.

(ii) It is straightforward to check that the set

{(u, u), (v, v), (w, w), (v, y), (y, v), (u,z), (z, u)}

constitutes a dominating set of G1 x G1. []

For any i/> 1 let G~ i) be an isomorphic copy of the graph Gl (where G~ i) = G1).

Label the vertices of the graphs G~ i) as it is shown in Fig. 2. Let Gn be the graph which

we obtain from the disjoin union of the graphs GI 1), ~1~(2), • • -, ~l~(n) with the addition of

edges {wi, ui+l} , 1 <~ i < n (see Fig. 2).

Theorem 2.2. (i) y(G.) = 3n.

(ii) y(G, x G,) ~< 7n 2.

Proof. (i) By Lemma 2.1(i) the domination number of any graph G] i) does not depend

on the vertices ui and wi. Therefore, each G~ i) must contain at least 3 vertices of a

minimum dominating set of Gn.

(ii) Let G~' -- Ui~l GI i) be the disjoint union of the graphs G~ 0. Clearly, G" × G' n is

a (proper) subgraph of the product G n × Gn and hence

7(Gn x Gn) ~< ~'(G~,, x G~n).

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S. Klav~ar, B. Zmazek/Discrete Mathematics 156 (1996) 243-246

The graph G~' × Gn / consists of n 2 connected components which are all isomorphic to

the product G1 x G1. Using Lemma 2.1(ii) we thus infer

7(Gn r x Gn' ) ---- n27(G1 × G1) ~< 7n 2,

which completes the proof. []

We can now state the main result of this note.

Corollary 2.3. For any k >>. 0 there exists a 9raph G such that

),(G × G) ~< y(G) 2 - k.

Proof. Set n = [v/~]. Theorem 2.2 immediately gives

~( G n x an) ~ ~(Gn) 2 - 2n 2

which in turn implies that

~:(Gn × Gn) <~ y(Gn) 2 - k. []

To conclude we wish to add that the above result indicates that domination problems

are not only interesting on the Cartesian product graphs but also on the direct product

graphs.

References

[1] T. Chang and E. Clark, The domination numbers of the 5 × n and 6 x n grid graphs, J. Graph Theory

17 (1993) 81-107.

[2] M. EI-Zahar and C.M. Pareek, Domination number of products of graphs, Ars Combin. 31 (1991)

223-227.

[3] R.J. Faudree and R.H. Schelp, The domination number for the product of graphs, Congr. Numer. 79

(1990) 29-33.

[4] D.C. Fisher, Domination, fractional domination, 2-packing, and graph products, SIAM J. Discrete Math.

7 (1994) 493-498.

[5] S. Gravier and A. Khelladi, On the domination number of cross products of graphs, Discrete Math. 145

(1995) 273-277.

[6] B.L. Hartnell and D.F. Rail, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96.

[7] M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs: I, Ars Combin. 18

(1983) 33-44.

[8] M.S. Jacobson and L.F. Kinch, On the domination of the products of graphs II: trees, J. Graph Theory

10 (1986) 97-106.

[9] S. Klav~ar and N. Seifter, Dominating Cartesian products of cycles, Discrete Appl. Math. 59 (1995)

129-136.

[10] R.J. Nowakowski and D. Rail, Associative graph products and their independence, domination and

coloring numbers, manuscript, June 1993.

[11] V.G. Vizing, The Cartesian product of graphs, Vyc. Sis. 9 (1963) 30-43.